i20 Mr Brill, On the Application of Quaternions [Nov. 24, 



(3) Note on the Application of Quaternions to the Discussion 

 of Laplace's Equation. By J. Brill, M.A., St John's College. 



1. The object of the following communication is to obtain, as 

 far as is possible, with the aid of the Calculus of Quaternions, a 

 theory for the three-dimensional form of Laplace's Equation 

 analogous to the well-known theory of Conjugate Functions, which 

 has proved of so much service in the treatment of the two-dimen- 

 sional form. 



The two related solutions of the two-dimensional theory are 

 replaced in the three-dimensional theory, not by three, but by 

 four related solutions. Thus if a, ft, 7, & be four quantities con- 

 nected by the equations 



d8_d_y_dft 



dx dy dz ' 



98 da dy 



dy dz dx ' 



d8_dft_da 



dz dx dy ' 



dx dy dz ' 



then it is easily verified that a, ft, 7, 8 are severally solutions of 

 Laplace's Equation. Moreover, if we interpret 8 as the velocity 

 potential of a case of irrotational fluid motion, then a, ft, 7 are 

 the components of the vector potential which occupies a similar 

 place in the three-dimensional theory to that occupied by the 

 current function in the two-dimensional theory. 



If we now write 



r = — 8 + ia +jft + k<y, 

 we have 



_ _/ck dft dy\ .(_<$ d J_d0 

 \dx dy dz) \ dx dy dz 



.( dB , da dy\ '( d8 dft da\ n 

 + A-dy + d-z-i) +k [-dz + £-dy)-° (1 > 



Further, let 



p = — 2ix + jy + kz, 



a = ix — 2jy + kz, 

 t = ix + jy — 2kz. 

 Then we have 



Vp = Vo- = Vt = 0, 



